Theorem ttcwf2 37491

Description: If a transitive closure class is a set, then it is well-founded, assuming Regularity. (Contributed by Matthew House, 6-Apr-2026.)

Assertion

Ref	Expression
ttcwf2	⊢ (TC+ 𝐴 ∈ V ↔ TC+ 𝐴 ∈ ∪ (𝑅1 “ On))

Proof of Theorem ttcwf2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 488	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∧ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅) → 𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)))
2	1	eldifad 4054	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∧ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅) → 𝑥 ∈ TC+ 𝐴)
3		ttctr2 37460	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ TC+ 𝐴 → 𝑥 ⊆ TC+ 𝐴)
4	2, 3	syl 18	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∧ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅) → 𝑥 ⊆ TC+ 𝐴)
5		dfss2 4097	. . . . . . . . . . . 12 ⊢ (𝑥 ⊆ TC+ 𝐴 ↔ (𝑥 ∩ TC+ 𝐴) = 𝑥)
6	4, 5	sylib 221	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∧ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅) → (𝑥 ∩ TC+ 𝐴) = 𝑥)
7		inssdif0 4325	. . . . . . . . . . . 12 ⊢ ((𝑥 ∩ TC+ 𝐴) ⊆ ∪ (𝑅1 “ On) ↔ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅)
8	7	bilanri 512	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∧ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅) → (𝑥 ∩ TC+ 𝐴) ⊆ ∪ (𝑅1 “ On))
9	6, 8	eqsstrrd 3948	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∧ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅) → 𝑥 ⊆ ∪ (𝑅1 “ On))
10		vex 3458	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
11	10	r1elss 9841	. . . . . . . . . 10 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) ↔ 𝑥 ⊆ ∪ (𝑅1 “ On))
12	9, 11	sylibr 237	. . . . . . . . 9 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∧ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅) → 𝑥 ∈ ∪ (𝑅1 “ On))
13	1	eldifbd 4055	. . . . . . . . 9 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∧ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅) → ¬ 𝑥 ∈ ∪ (𝑅1 “ On))
14	12, 13	pm2.65da 829	. . . . . . . 8 ⊢ (𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) → ¬ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅)
15	14	nrex 3094	. . . . . . 7 ⊢ ¬ ∃𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))(𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅
16	15	a1i 11	. . . . . 6 ⊢ (TC+ 𝐴 ∈ V → ¬ ∃𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))(𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅)
17		difexg 5295	. . . . . . 7 ⊢ (TC+ 𝐴 ∈ V → (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∈ V)
18		zfreg 9617	. . . . . . 7 ⊢ (((TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∈ V ∧ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ≠ ∅) → ∃𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))(𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅)
19	17, 18	sylan 592	. . . . . 6 ⊢ ((TC+ 𝐴 ∈ V ∧ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ≠ ∅) → ∃𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))(𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅)
20	16, 19	mtand 828	. . . . 5 ⊢ (TC+ 𝐴 ∈ V → ¬ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ≠ ∅)
21		nne 2963	. . . . 5 ⊢ (¬ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ≠ ∅ ↔ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) = ∅)
22	20, 21	sylib 221	. . . 4 ⊢ (TC+ 𝐴 ∈ V → (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) = ∅)
23		ssdif0 4317	. . . 4 ⊢ (TC+ 𝐴 ⊆ ∪ (𝑅1 “ On) ↔ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) = ∅)
24	22, 23	sylibr 237	. . 3 ⊢ (TC+ 𝐴 ∈ V → TC+ 𝐴 ⊆ ∪ (𝑅1 “ On))
25		eleq1 2852	. . . . 5 ⊢ (𝑥 = TC+ 𝐴 → (𝑥 ∈ ∪ (𝑅1 “ On) ↔ TC+ 𝐴 ∈ ∪ (𝑅1 “ On)))
26		sseq1 3938	. . . . 5 ⊢ (𝑥 = TC+ 𝐴 → (𝑥 ⊆ ∪ (𝑅1 “ On) ↔ TC+ 𝐴 ⊆ ∪ (𝑅1 “ On)))
27	25, 26	bibi12d 348	. . . 4 ⊢ (𝑥 = TC+ 𝐴 → ((𝑥 ∈ ∪ (𝑅1 “ On) ↔ 𝑥 ⊆ ∪ (𝑅1 “ On)) ↔ (TC+ 𝐴 ∈ ∪ (𝑅1 “ On) ↔ TC+ 𝐴 ⊆ ∪ (𝑅1 “ On))))
28	27, 11	vtoclg 3521	. . 3 ⊢ (TC+ 𝐴 ∈ V → (TC+ 𝐴 ∈ ∪ (𝑅1 “ On) ↔ TC+ 𝐴 ⊆ ∪ (𝑅1 “ On)))
29	24, 28	mpbird 260	. 2 ⊢ (TC+ 𝐴 ∈ V → TC+ 𝐴 ∈ ∪ (𝑅1 “ On))
30		elex 3475	. 2 ⊢ (TC+ 𝐴 ∈ ∪ (𝑅1 “ On) → TC+ 𝐴 ∈ V)
31	29, 30	impbii 212	1 ⊢ (TC+ 𝐴 ∈ V ↔ TC+ 𝐴 ∈ ∪ (𝑅1 “ On))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900   ∖ cdif 4048   ∩ cin 4092  ∅c0 4281  ∪ cuni 4870   “ cima 5658  Oncon0 6380  𝑅₁cr1 9793  TC+ cttc 37452

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-reg 9613

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7446  df-om 7902  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-r1 9795  df-ttc 37453

This theorem is used by:  ttcwf3  37492